1. **State the problem:** We have a rectangular piece of aluminum with length $x+3$ cm and width $x$ cm. Four squares of side length 2 cm are cut from each corner, and the sides are folded up to form an open box with height 2 cm. The volume of this box is 176 cubic centimeters. 2. **Write the volume formula:** The volume $V$ of a box is given by: $$V = \text{length} \times \text{width} \times \text{height}$$ 3. **Determine the dimensions of the box after cuts:** Since squares of side 2 cm are cut from each corner, the length and width of the base of the box become: $$\text{length}_{box} = (x+3) - 2 \times 2 = x + 3 - 4 = x - 1$$ $$\text{width}_{box} = x - 4$$ The height is the side length of the cut squares, which is 2 cm. 4. **Write the volume equation:** Given volume $V = 176$ cm$^3$, we have: $$176 = (x - 1)(x - 4)(2)$$ 5. **Simplify the equation:** Divide both sides by 2: $$\frac{176}{\cancel{2}} = (x - 1)(x - 4)\cancel{(2)}$$ $$88 = (x - 1)(x - 4)$$ 6. **Expand the right side:** $$88 = x^2 - 4x - x + 4 = x^2 - 5x + 4$$ 7. **Write the quadratic equation:** Bring all terms to one side: $$x^2 - 5x + 4 - 88 = 0$$ $$x^2 - 5x - 84 = 0$$ 8. **Solve the quadratic equation:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-5$, $c=-84$. Calculate the discriminant: $$\Delta = (-5)^2 - 4(1)(-84) = 25 + 336 = 361$$ Calculate the roots: $$x = \frac{5 \pm \sqrt{361}}{2} = \frac{5 \pm 19}{2}$$ Two possible solutions: $$x = \frac{5 + 19}{2} = 12$$ $$x = \frac{5 - 19}{2} = -7$$ Since $x$ represents a length, it must be positive, so $x = 12$ cm. 9. **Find the original dimensions:** Length: $$x + 3 = 12 + 3 = 15 \text{ cm}$$ Width: $$x = 12 \text{ cm}$$ **Final answers:** - Quadratic equation: $$x^2 - 5x - 84 = 0$$ - Original length: 15 cm - Original width: 12 cm